How to Derive the Density Difference in Van Der Waals Phase Transition?

[Diracobama2181]

Homework Statement

Show that $$\rho_{gas}-\rho_{liquid}\propto |T_C-T|^\frac{1}{2}$$.

Relevant Equations

$$P=\frac{\rho RT}{1-\rho \beta}-\alpha \rho^2$$

Not sure where to actually start. Do I need to do a virial expansion? Any tips on on where to start would be greatly appreciated.

 

[Chestermiller]

Start by writing down alpha and beta in terms of TC and PC.

 

[Abhishek11235]

Rearrange the given equation for T. Now,at critical point density is infinite. So, use limiting process for density.

 

[Diracobama2181]

Rearrange the given equation for T. Now,at critical point density is infinite. So, use limiting process for density.

Would T and P be different for $$\rho_{gas}$$ and $$\rho_{liquid}$$?
Right now, after rearranging, I get
$$T=\frac{P-\rho \beta P+\alpha \rho^2-\alpha \beta \rho^3}{R\rho}$$
which gives
$$T=\frac{- \beta P+\alpha \rho-\alpha \beta \rho^2}{R}$$
when I let $$\rho$$ go to $$ \infty$$

 

[Abhishek11235]

which gives
$$T=\frac{- \beta P+\alpha \rho-\alpha \beta \rho^2}{R}$$
when I let $$\rho$$ go to $$ \infty$$

As it should(There is very exciting physical phenomenon related to this). Now you want to find relation between phase change and density. For this,you have to approach one temperature(The critical temperature)(Why?). Next,the pressure should be same(This should become clear if you P-T graph of phase change relationship)

 

[Fred Wright]

Please see the excellent discussion here, http://www.damtp.cam.ac.uk/user/tong/statphys/five.pdf, and how proportionality 5.9 is derived.

 

[WidomFisherLine]

Homework Statement:: Show that $$\rho_{gas}-\rho_{liquid}\propto |T_C-T|^\frac{1}{2}$$.
Homework Equations:: $$P=\frac{\rho RT}{1-\rho \beta}-\alpha \rho^2$$

Not sure where to actually start. Do I need to do a virial expansion? Any tips on on where to start would be greatly appreciated.

Use the Widom Insertion Method